Suppose I have a family of objects $\{A_i\}$ which have a product $({\Pi}A_i, \{\rho_i\}) $ in a category $\mathcal{C}$.
Is there a product in $\mathcal{C}$ for every nonempty subset of $\{A_i\}$? If not, what is a counter-example?
Can this be done right from the definition of categorical products?
Not necessarily.
For example, in a poset, viewed as a category, the product of a family of objects is their greatest lower bound. Now it's quite possible to have a poset $P$ and sets $X\subseteq Y\subseteq P$ such that $Y$ has a greatest lower bound but $X$ does not.
The simplest example has $|X|=2$, $|Y|=3$, and $|P|=4$. Take $P=\{a,b,c,d\}$, with $a,b\leq c,d$. The greatest lower bound of $Y=\{a,c,d\}$ is $a$, but $X=\{c,d\}$ has no greatest lower bound, since the two lower bounds for $X$ are $a$ and $b$, which are incomparable.